Biomedical Engineering Reference
In-Depth Information
where the gradient of
U
is discussed inAppendixAand is given in Cartesian coordinates by
∂
U
∂
x
e
x
+
∂
U
∂
y
e
y
+
∂
U
∂
z
e
z
grad
U
=
(1.15)
Here
e
x
,
e
y
and
e
z
are unit vectors pointing along the Cartesian axes.
1.9 Electric Multipoles
We can define exactly an array of point charges by listing the magnitudes of the charges,
together with their position vectors. If we then wish to calculate (say) the force between one
array of charges and another, we simply apply Coulomb's law (Equation (1.3)) repeatedly
to each pair of charges. Equation (1.3) is exact, and can be easily extended to cover the
case of continuous charge distributions.
For many purposes, it proves more profitable to describe a charge distribution in terms
of certain quantities called the
electric moments
. We can then discuss the interaction of one
charge distribution with another in terms of the interactions between the electric moments.
Consider first a pair of equal and opposite point charges,
Q
separated by
distance
R
(Figure 1.8). This pair of charges is usually said to form an
electric dipole
of
magnitude
QR
. In fact, electric dipoles are vector quantities and amore rigorous definition is
+
Q
and
−
p
e
=
Q
R
(1.16)
where the vector
R
points from the negative charge to the positive charge.
+
Q
R
-
Q
Figure 1.8
Simple electric dipole
We sometimes have to concern ourselves with a more general definition, one relating to
an arbitrary array of charges such as that shown in Figure 1.9. We have four point charges:
Q
1
whose position vector is
R
1
,
Q
2
whose position vector is
R
2
,
Q
3
whose position vector
is
R
3
and
Q
4
whose position vector is
R
4
. We define the
electric dipole moment
p
e
of these
four charges as
4
p
e
=
Q
i
R
i
i
=
1
It is a vector quantity with
x
,
y
and
z
Cartesian components
4
4
4
Q
i
X
i
,
Q
i
Y
i
and
Q
i
Z
i
i
=
1
i
=
1
i
=
1
